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MATH 125 Calculus II

2021-08-17

Contents

Integrals

Indefinite integrals

Description	Equations
Indefinite integral (antiderivative)	$F(x) = \displaystyle\int f(x) \ dx \newline F'(x) = f(x)$
Antiderivative as a family of functions (Plus $C$ !)	If $F$ is an antiderivative of $f$ , $C$ is a constant, then the most general antiderivative is $F(x) + C$

Table of indefinite integrals

Function $f(x)$	Antiderivative $F(x)$	Function $f(x)$	Antiderivative $F(x)$
$x^n$	$\dfrac{x^{n+1}}{n+1}+C$	$\dfrac{1}{x}$	$\ln\lvert x \rvert + C$
$e^x$	$e^x + C$	$b^x$	$\dfrac{b^x}{\ln b} + C$
$\sin x$	$-\cos x + C$	$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$	$csc^2 x$	$-\cot x + C$
$\sec x\tan x$	$\sec x + C$	$\csc x\cot x$	$-\csc x + C$
$\dfrac{1}{x^2 + a^2}$	$\dfrac{1}{a}\arctan \left(\dfrac{x}{a}\right) + C$	$\dfrac{1}{\sqrt{a^2-x^2}}$	$\arcsin \left(\dfrac{x}{a}\right) + C$

Definite integrals as Riemann sums

Description	Equations
Area	$A = \lim\limits_{n\to\infin} R_{n} = \lim\limits_{n\to\infin} \sum\limits_{i=1}^{n}f(x_i)\Delta x$
Definite integral	$\int_{a}^{b} f(x) \ dx = \lim\limits_{n\to\infin} \sum\limits_{i=1}^{n}f(x_i^*)\Delta x$
Operational definition of definite integral as Riemann sum	$\int_a^b f(x) \ dx = \lim\limits_{n\to\infin} \sum\limits_{i=1}^n f(x_i)\Delta x$ $\Delta x = \frac{b-a}{n} \newline x_i = a+i\Delta x$
Sums of powers of positive integers	$\sum\limits_{i=1}^{n}i = \frac{n(n+1)}{2} \newline \sum\limits_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6} \newline \sum\limits_{i=1}^{n}i^3 = \left[ \frac{n(n+1)}{2} \right]^2$
Properties of summation	$\sum\limits_{i=1}^{n}c = nc \newline \sum\limits_{i=1}^{n}ca_i = c\sum\limits_{i=1}^{n}a_i \newline \sum\limits_{i=1}^{n}(a_i \pm b_i) = \sum\limits_{i=1}^{n}a_i \pm \sum\limits_{i=1}^{n}b_i$

Properties of definite integrals

Description	Equations
Reversing the bounds changes the sign of definite integrals	$\int_a^b f(x) \ dx = -\int_b^a f(x) \ dx$
Definite integral is zero if upper and lower bounds are the same	$\int_a^a f(x) \ dx = 0$
Definite integrals of constant	$\int_a^b c \ dx = c(b-a)$
Addition and subtraction of definite integrals	$\int_a^b [f(x) \pm g(x)] \ dx \newline = \int_a^b f(x) \ dx \pm \int_a^b g(x) \ dx$
Constant multiple of definite integrals	$\int_a^b cf(x) \ dx = c\int_a^b f(x) \ dx$
Comparison properties of definite integrals	If $f(x) \ge 0$ for $x\in[a,b]$ , then $\int_a^b f(x) \ dx \ge 0$
Comparison properties of definite integrals	If $f(x) \ge g(x)$ for $x\in[a,b]$ , then $\int_a^b f(x) \ dx \ge \int_a^b g(x) \ dx$
Comparison properties of definite integrals	If $m \le f(x) \le M$ for $x\in[a,b]$ , then $m(b-a) \le \int_a^b f(x) \ dx \le M(b-a)$

Fundamental theorem of calculus

Description	Equations
Fundamental theorem of calculus I ( $f$ is continuous on $[a,b]$ )	$g(x) = \displaystyle\int_a^x f(t) \ dt \newline g'(x) = f(x)$
Fundamental theorem of calculus II ( $f$ is continuous on $[a,b]$ )	$\displaystyle\int_a^b f(x) \ dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$
Net change theorem The integral of a rate of change is the net change	$\displaystyle\int_a^b F'(x) \ dx = F(b) - F(a)$

Substitution rule

Description	Equations
Substitution rule (u-substitution) $u \equiv g(x)$	$\displaystyle\int f(g(x)) g'(x) \ dx = \int f(u) \ du$
Substitution rule for definite integrals $u \equiv g(x)$	$\displaystyle\int_a^b f(g(x))g'(x) \ dx = \int_{g(a)}^{g(b)} f(u) \ du$
Integrals of even functions	$\int_{-a}^a f(x) \ dx = 2 \int_{0}^a f(x) \ dx$
Integrals of odd functions	$\int_{-a}^a f(x) \ dx = 0$

Techniques of Integration

Integration by parts

Description	Equations
Integration by parts	$\int f(x)g'(x) \ dx \newline = f(x)g(x) - \int g(x)f'(x) \ dx$
Integration by parts	$\int u \ dv = uv - \int v \ du$
Integration by parts for definite integrals	$\int_a^b fg' \ dx = [fg]_a^b - \int_a^b f’g \ dx$

Approximating integrals

Description	Equations
Midpoint rule	$\int_a^b f(x) \ dx \approx \sum\limits_{i=1}^n f(\bar{x}_i)\Delta x$ $\Delta x = \frac{b-a}{n} \newline \bar{x}_i = \frac{1}{2}(x_{i-1}+x_i)$
Error bound for midpoint rule	$\lvert E_M \rvert \le \dfrac{K(b-a)^3}{24n^2}$
Trapezoidal rule	$\int_a^b f(x) \ dx \approx \frac{1}{2}\Delta x [f(x_0) + 2f(x_1) + … + 2f(x_{n-1}) + f(x_n)]$ $\Delta x = \frac{b-a}{n} \newline x_i = a + i\Delta x$
Error bound for trapezoidal rule	$\lvert E_T \rvert \le \dfrac{K(b-a)^3}{12n^2}$
Simpson’s rule	$\int_a^b f(x) \ dx \approx \frac{1}{3}\Delta x [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + … + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$ $\Delta x = \frac{b-a}{n}$ , n is even
Error bound for Simpson’s rule	$\lvert E_S \rvert \le \dfrac{K(b-a)^5}{180n^4}$

Trigonometric integrals

Description	Equations
Integral of odd power of cosine $(u = \sin x)$	$\int \sin^m(x)\cos^{2k+1}(x) \ dx \newline = \int \sin^m(x) [\cos^2 (x)]^k \ dx \newline = \int \sin^m(x)[1-\sin^2(x)]^k \ dx$
Integral of odd power of sine $(u = \cos x)$	$\int \sin^{2k+1}(x)\cos^{n}(x) \ dx \newline = \int [\sin^2 (x)]^k \cos^n(x) \sin(x) \ dx \newline = \int [1-\cos^2(x)]^k \cos^n(x) \sin(x) \ dx$
Integral of even power of sine and cosine use trig identities	$\sin^2(x) = \frac{1}{2}(1-\cos(2x)) \newline \cos^2(x) = \frac{1}{2}(1+\cos(2x)) \newline \sin(x)\cos(x) = \frac{1}{2}\sin(2x)$
Integral of even power of secant $(u = \tan x)$	$\int \tan^m(x)\sec^{2k}(x) \ dx \newline = \int \tan^m(x)[\sec^2(x)]^{k-1}\sec^2(x) \ dx \newline = \int \tan^m(x)[1+\tan^2(x)]^{k-1}\sec^2(x) \ dx$
Integral of odd power of tangent $(u = \sec x)$	$\int tan^{2k+1}(x)\sec^n(x) \ dx \newline = \int[\tan^2(x)]^k\sec^{n-1}(x)\sec(x)\tan(x) \ dx \newline = \int [\sec^2(x)-1]^k\sec^{n-1}(x)\sec(x)\tan(x) \ dx$
Trig identity for solving $\int \sin(mx)\cos(nx) \ dx$	$\sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)]$
Trig identity for solving $\int \sin(mx)\sin(nx) \ dx$	$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$
Trig identity for solving $\int \cos(mx)\cos(nx) \ dx$	$\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$

Trigonometric substitution

Expression	Substitution	Trigonometric Identity
$\sqrt{a^2 - x^2}$	$x = a\sin\theta$	$1 - \sin^2\theta = \cos^2\theta$
$\sqrt{a^2 + x^2}$	$x = a\tan\theta$	$1+\tan^2\theta = \sec^2\theta$
$\sqrt{x^2 - a^2}$	$x = a\sec\theta$	$\sec^2\theta-1 = \tan^2\theta$

Improper integrals

Description	Equations
Improper integrals with single one-side infinite intervals	$\int_a^\infin f(x) \ dx = \lim\limits_{t\to\infin}\int_a^t f(x) \ dx \newline \int_{-\infin}^b f(x) \ dx = \lim\limits_{t\to-\infin}\int_t^b f(x) \ dx$
Improper integrals with single two-side infinite intervals	$\int_{-\infin}^\infin f(x) \ dx \newline = \int_{-\infin}^a f(x) \ dx + \int_a^\infin f(x) \ dx$
Convergence and divergence of improper integrals of power functions	$\displaystyle\int_1^\infin \dfrac{1}{x^p} \ dx$ convergent if $p>1$ divergent if $p \le 1$
Improper integrals with discontinuous integrand on one side	$\int_a^b f(x) \ dx = \lim\limits_{t\to b^-}\int_a^t f(x) \ dx \newline \int_a^b f(x) \ dx = \lim\limits_{t\to a^+}\int_t^b f(x) \ dx$
Improper integrals with discontinuous integrand in the middle $c$	$\int_a^b f(x) \ dx = \int_a^c f(x) \ dx + \int_c^b f(x) \ dx$
Comparison theorem $(f(x) \ge g(x) \ge 0, x \ge a)$	(a) If $\int_a^\infin f(x) \ dx$ is convergent, then $\int_a^\infin g(x) \ dx$ is convergent. (b) If $\int_a^\infin g(x) \ dx$ is divergent, then $\int_a^\infin f(x) \ dx$ is divergent.

Applications of Integration

Description	Equations
Areas between curves	$A = \int_a^b [f(x) - g(x)] \ dx$
Volume by method of disks and washers	$V = \int_a^b A(x) \ dx$
Volume by method of cylindrical shells (rotating about y-axis)	$V = \int_a^b 2\pi x f(x) \ dx$
Average value of a function	$\bar{f} = \frac{1}{b-a}\int_a^b f(x) \ dx$
The mean value theorem of integrals	If $f$ is continuous on $[a,b]$ , then there exists $c\in[a,b]$ such that $f(c) = \bar{f} = \frac{1}{b-a}\int_a^b f(x) \ dx$ , $\int_a^b f(x) \ dx = f(c)(b-a)$
Arc length formula	$L = \int_a^b \sqrt{1+[f'(x)]^2} \ dx$
Arc length function	$s(x) = \int_a^x \sqrt{1+[f'(t)]^2} \ dt$
Surface area of surface of resolution about x-axis	$S = \int_a^b 2\pi f(x)\sqrt{1 + [f'(x)]^2} \ dx$